Given a circle with centre O and chords AB, PQ and XY. Points P, Q, and O are collinear and radius of a circle is 6 cm. Then mark the correct option.
(1 Point)
AB=XY = 3 cm
AB = 6 cm = XY
PQ= 6 cm
PQ= 12 cm
Answers
Answer:
Given a circle with centre O and chords AB, PQ and XY. Points P, Q, and O are collinear and radius of a circle is 6 cm. Then mark the correct option.
(1 Point)
AB=XY = 3 cm1) It is given that seg RM and seg RN are tangent segments touching the circle at M and N, respectively.
∴ ∠OMR = ∠ONR = 90º (Tangent at any point of a circle is perpendicular to the radius throught the point of contact)In ∆OMR and ∆ONR,
seg MR = seg NR (Tangent segments drawn from an external point to a circle are congruent)
seg OM = seg ON (Radii of the same circle)
seg OR = seg OR (Common)
∴ ∆OMR ≌ ∆ONR (SSS congruence criterion)
⇒ ∠ORM = ∠ORN (CPCT)
Also, ∠MOR = ∠NOR (CPCT)
∴ seg OR bisects ∠MRN and ∠MON.1) It is given that seg RM and seg RN are tangent segments touching the circle at M and N, respectively.
∴ ∠OMR = ∠ONR = 90º (Tangent at any point of a circle is perpendicular to the radius throught the point of contact)
OM = 5 cm and OR = 10 cm
In right ∆OMR,
OR2=OM2+MR2⇒MR=OR2−OM2−−−−−−−−−−√ ⇒MR=102−52−−−−−−−√⇒MR=100−25−−−−−−−√=75−−√=53–√ cm
Tangent segments drawn from an external point to a circle are congruent.
∴ MR = NR = 53–√ cm
(2) In right ∆OMR,
tan∠MRO=OMMR⇒tan∠MRO=5 cm53√ cm=13√⇒tan∠MRO=tan30°⇒∠MRO=30°
Thus, the measure of ∠MRO is 30º.
Similarly, ∠NRO = 30º
(3) ∠MRN = ∠MRO + ∠NRO = 30º + 30º = 60º
Thus, the measure of ∠MRN is 60º.
OM = 5 cm and OR = 10 cm
In right ∆OMR,
OR2=OM2+MR2⇒MR=OR2−OM2−−−−−−−−−−√ ⇒MR=102−52−−−−−−−√⇒MR=100−25−−−−−−−√=75−−√=53–√ cm
Tangent segments drawn from an external point to a circle are congruent.
∴ MR = NR = 53–√ cm
(2) In right ∆OMR,
tan∠MRO=OMMR⇒tan∠MRO=5 cm53√ cm=13√⇒tan∠MRO=tan30°⇒∠MRO=30°
Thus, the measure of ∠MRO is 30º.
Similarly, ∠NRO = 30º
(3) ∠MRN = ∠MRO + ∠NRO = 30º + 30º = 60º
Thus, the measure of ∠MRN is 60º.
AB = 6 cm = XY
PQ= 6 cm
PQ= 12 cm